# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.596 + 0.802i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.889 − 2.68i)2-s + (−6.41 + 4.77i)4-s + 14.9i·5-s − 30.0i·7-s + (18.5 + 12.9i)8-s + (40.0 − 13.2i)10-s + 55.9·11-s + 57.4·13-s + (−80.6 + 26.7i)14-s + (18.3 − 61.2i)16-s − 29.2i·17-s + 0.709i·19-s + (−71.2 − 95.7i)20-s + (−49.7 − 150. i)22-s + 48.0·23-s + ⋯
 L(s)  = 1 + (−0.314 − 0.949i)2-s + (−0.802 + 0.596i)4-s + 1.33i·5-s − 1.62i·7-s + (0.818 + 0.573i)8-s + (1.26 − 0.419i)10-s + 1.53·11-s + 1.22·13-s + (−1.54 + 0.510i)14-s + (0.287 − 0.957i)16-s − 0.417i·17-s + 0.00856i·19-s + (−0.796 − 1.07i)20-s + (−0.482 − 1.45i)22-s + 0.435·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.596 + 0.802i$ motivic weight = $$3$$ character : $\chi_{108} (107, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :3/2),\ 0.596 + 0.802i)$$ $$L(2)$$ $$\approx$$ $$1.23071 - 0.618331i$$ $$L(\frac12)$$ $$\approx$$ $$1.23071 - 0.618331i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (0.889 + 2.68i)T$$
3 $$1$$
good5 $$1 - 14.9iT - 125T^{2}$$
7 $$1 + 30.0iT - 343T^{2}$$
11 $$1 - 55.9T + 1.33e3T^{2}$$
13 $$1 - 57.4T + 2.19e3T^{2}$$
17 $$1 + 29.2iT - 4.91e3T^{2}$$
19 $$1 - 0.709iT - 6.85e3T^{2}$$
23 $$1 - 48.0T + 1.21e4T^{2}$$
29 $$1 - 172. iT - 2.43e4T^{2}$$
31 $$1 + 45.2iT - 2.97e4T^{2}$$
37 $$1 - 248.T + 5.06e4T^{2}$$
41 $$1 + 51.3iT - 6.89e4T^{2}$$
43 $$1 + 19.9iT - 7.95e4T^{2}$$
47 $$1 + 10.8T + 1.03e5T^{2}$$
53 $$1 - 37.0iT - 1.48e5T^{2}$$
59 $$1 - 411.T + 2.05e5T^{2}$$
61 $$1 + 308.T + 2.26e5T^{2}$$
67 $$1 + 113. iT - 3.00e5T^{2}$$
71 $$1 + 1.13e3T + 3.57e5T^{2}$$
73 $$1 - 728.T + 3.89e5T^{2}$$
79 $$1 + 487. iT - 4.93e5T^{2}$$
83 $$1 + 1.16e3T + 5.71e5T^{2}$$
89 $$1 + 1.19e3iT - 7.04e5T^{2}$$
97 $$1 - 624.T + 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}